Credal network

Credal networks are probabilistic graphical models based on imprecise probability. Credal networks can be regarded as an extension of Bayesian networks, where credal sets replace probability mass functions in the specification of the local models for the network variables given their parents. As a Bayesian network defines a joint probability mass function over its variables, a credal network defines a joint credal set. The way this credal set is defined depends on the particular notion of independence for imprecise probability adopted. Most of the research on credal networks focused on the case of strong independence. Given strong independence the joint credal set associated to a credal network is called its strong extension. Let ${\displaystyle (X_{1},\ldots ,X_{n})}$ denote a collection of categorical variables and ${\displaystyle G}$. If ${\displaystyle K(X_{i}\mid \pi _{i})}$ is, for each ${\displaystyle i=1,\ldots ,n}$, a conditional credal set over ${\displaystyle X_{i}}$, then the strong extension of a credal network is defined as follows:

${\displaystyle K(X_{1},\ldots ,X_{n})=\mathrm {CH} \{P(X_{1},\ldots ,X_{n}):P(x_{1},\ldots ,x_{n})=\prod _{i=1}^{n}P(x_{i}\mid \pi _{i}),P(X_{i}\mid \pi _{i})\in K(X_{i}\mid \pi _{i})\}}$

where ${\displaystyle \mathrm {CH} }$ denote the convex hull.